有限差分法应用于具有复杂外形的网格时需要进行坐标变换，在此过程中经常会引入坐标变换诱导误差。在柱坐标系下使用均匀网格进行均匀流场计算，计算结果表明，即使物理坐标对计算坐标的变换函数连续可导、计算过程中坐标变换系数直接采用准确的解析式、网格完全正交并且充分光滑，也无法避免坐标变换诱导误差。理论分析表明，产生坐标变换诱导误差的机理是笛卡尔坐标系下的守恒型欧拉方程变换至贴体坐标系下后增加了源项。针对该问题，目前国内外学者通常采用几何守恒律，构建与差分格式相匹配的坐标变换系数计算方法来消除源项。本文介绍了从包含源项的离散等价方程基础上直接进行离散的新算法，在此基础上针对非等距网格条件下MUSCL类格式重构过程进行误差分析，理论推导表明重构中需要考虑非等距插值公式的影响系数，将变量转换至计算空间内进行MUSCL重构才能保证该过程具有均匀网格下的插值精度。通过理论分析及数值实验证明新算法对于均匀流场完全不会引入坐标变换误差。
Coordinate transformation is required when the finite difference method is applied to the mesh with complex geometries, and the errors induced by the coordinate transformation are often introduced in this process. These errors are proved to be inevitable in the uniform flow field calculation with uniform grids in cylindrical coordinate systems, even if the transformation function of the physical coordinates to the calculated coordinates is continuously derivable, or the coordinate transformation coefficients in the calculation process are calculated by the accurate analytical formula, or the grid is completely orthogonal and fully smooth. Theoretical analysis shows the mechanism of the coordinate transformation induced errors:when the conservative Euler equation is transformed from the Cartesian coordinate system to the body fitted coordinate system, a source term is added. Currently, scholars usually use the geometric conservation law to construct a method based on coordinate transformation coefficients, which are matched with the format of the finite difference, to eliminate the source term. In this work, we introduce a new algorithm that processes the direct discretization from the discrete equivalent functions including the source term. Based on the above new algorithm, error analysis is carried out for the reconstruction process of MUSCL format under non-equidistant grid conditions. Theoretical derivation shows that the influence coefficient of the non-equidistant interpolation formula needs to be considered in reconstruction, only when the variables are transformed into the computational space for MUSCL reconstruction can the interpolation accuracy be guaranteed under uniform grid. Our theoretical analysis and numerical experiments have proven that this algorithm will not introduce coordinate transformation errors to the uniform flow field calculations.
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